Two particles of equal mass m are at the vertices of the...

Two particles of equal mass $m$ are at the vertices of the base of an equilateral triangle. The triangle's center of mass is midway between the base and the third vertex. What's the mass at the third vertex?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Center of Mass

The center of mass of a system is the point representing the average location of the total mass. It is calculated by taking the weighted average of the positions of all particles, with each particle’s mass contributing according to its value.

Moment Balance

The moment (or torque) balance method involves ensuring that the sum of the products of each mass by its distance (or position vector) from a chosen reference point equals the corresponding product for the entire system. This balance is crucial when determining the location of the center of mass.

Geometric Considerations

In problems where masses are arranged in specific geometric configurations, like a triangle, the spatial distribution provided by the shape's geometry is key. The known positions of the masses combined with geometrical properties allow one to establish equations for the center of mass location.

Transcript

00:02 Okay, so in this question, we have two particles at the base of any collateral triangle, and we're told where the center of mass is, midway between the base and the third vertex.

00:14 So we want to get the mass at the third vertex.

00:17 So this is interesting.

00:18 So let's go ahead and draw it.

00:22 So mass m, mass m.

00:28 And then here is the third mass.

00:31 We'll call this m prime.

00:34 I'm, okay, and then we are told that the center of mass of this whole thing is like right in the center of here.

00:43 And so we want to get what m is.

00:47 So let's just kind of think about this problem first in broad strokes.

00:51 So if this mass was the same as these, the center of mass should be lower, but right, because there's more mass towards the bottom of the triangle.

01:01 But this mass, so this, therefore this must be bigger than.

01:05 The halfway point.

01:06 So let's just keep that in mind.

01:07 Hope we get that.

01:08 So now let's just do calculate a formula for the vertical or write a formula for the vertical center of mass.

01:13 So i'll call this the y direction.

01:16 So in general, the center of mass is going to be the sum of each mass times its position in the y direction divided by the total mass...